Degree of rational maps via specialization

TL;DR

This paper investigates how the degree of a rational map changes under specialization of coefficients, using classical Kronecker methods and developing the theory over Noetherian integral domains.

Contribution

It extends the classical theory of rational maps and their degrees to the setting of Noetherian integral domains, providing a new approach to specialization.

Findings

Degree behavior under specialization is characterized.

Develops a detailed theory of rational maps over Noetherian integral domains.

Provides a framework for analyzing rational maps via their graphs.

Abstract

One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian integral domain.